Finite Group Databasebeta

← #27309601

Subgroups — #27309601

Order 13568.

1894 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810534C211yesmaximal, characteristic
2C2×#16810514C211yesmaximal, characteristic
3C2×#16810513C212yesmaximal
4#20272986C211yesmaximal, characteristic, complemented
5#20272974C211yesmaximal, characteristic, complemented
6#20272957C211yesmaximal, characteristic
7#20272953C212yesmaximal
8#20272948C212yesmaximal, complemented
9#20272947C212yesmaximal, complemented
10C4×#13133124C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C53C211yesmaximal, characteristic, complemented
12C22×#11719919C2211yescharacteristic
13C22×#11719921C2211yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2211yescharacteristic
18C22×C4×Dic53C2212yes—
19#16810534C418yescomplemented
20#16810514—24no—
21#16810513—28no—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C23×Q8×C53C2211yescharacteristic
34C2×C4×Q8×C53—28no—
35C2×C4×Q8×C53—24no—
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2212yes—
38C22×C42×C53C2212yes—
39C22×C42×C53C2211yescharacteristic
40C24×D53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C2×#11719919—22no—
47C2×#11719919—24no—
48C2×#11719921—24no—
49C2×#11719921—22no—
50C2×#11719921—24no—
51C2×#11719921—24no—
52C2×#11719921—22no—
53C2×#11719921—22no—
54C2×#11719921—24no—
55C2×#11719921—22no—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—42no—
59C2×#11719921C2×C418yes—
60C2×#11719921C2×C414yes—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—42no—
64#13133124—24no—
65#13133124—24no—
66#13133124—22no—
67#13133124—22no—
68#13133124—22no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—42no—
73#13133124C2×C414yes—
74#13133124C2×C418yes—
75C22×Q8×C53—24no—
76C22×Q8×C53—22no—
77C22×Q8×C53—22no—
78C22×Q8×C53—24no—
79C22×Q8×C53C2×C414yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C4×Q8×C53—232no—
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53—24no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53D418yes—
91C2×C42×C53—24no—
92C2×C42×C53—24no—
93C2×C42×C53—24no—
94C2×C42×C53—24no—
95C2×C42×C53D414yes—
96C24×C53C2411yescharacteristic
97C23×D53—21no—
98C23×D53—22no—
99C23×D53—21no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—21no—
105C23×D53—21no—
106C23×D53—21no—
107C23×D53C22×C412yes—
108C2×Q8×C53—28no—
109C2×Q8×C53—28no—
110C2×Q8×C53—216no—
111C2×Q8×C53—28no—
112C2×Q8×C53—216no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53C22×C414yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D414yes—
134C22×C4×C53C2×D414yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D414yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C22×C4×C53#4114yes—
141C22×C4×C53#4114yes—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C414yes—
170C22×Dic53#4114yes—
171C4⋊C4×C53—28no—
172C4⋊C4×C53—216no—
173C42×C53—216no—
174C42×C53—28no—
175#11719921—216no—
176#11719921—28no—
177#11719921—48no—
178#11719921—44no—
179#11719921—44no—
180#11719919—44no—
181C4×Dic53—48no—
182C23×C53C23×C411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D412yes—
185C23×C53C2×#4112yes—
186C23×C53C2×#4111yescharacteristic
187C23×C53C2×#4112yes—
188C23×C53C2×#4112yes—
189C23×C53H(5,𝔽2)11yescharacteristic
190C23×C532−1+411yescharacteristic
191C23×C532−1+411yescharacteristic
192C23×C532−1+411yescharacteristic
193C22×D53—22no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—42no—
201C22×D53—41no—
202C22×D53—42no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216Q8×C53—232no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53C4×D414yes—
243C2×C4×C53C4×D418yes—
244C2×C4×C53#12014yes—
245C2×C4×C53#12414yes—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—42no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275#14188—531nomaximal, Sylow
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×C4×D412yes—
278C22×C53C4×#4112yes—
279C22×C53#51911yescharacteristic
280C22×C53#52011yescharacteristic
281C22×C53C2×#12011yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53#52512yes—
284C22×C53C2×#12411yescharacteristic
285C22×C53#52912yes—
286C22×C53#53211yescharacteristic
287C22×C53#53611yescharacteristic
288C22×C53#53711yescharacteristic
289C22×C53#54111yescharacteristic
290C22×C53#54211yescharacteristic
291C22×C53#54612yes—
292C22×C53#54712yes—
293C22×C53#54812yes—
294C22×C53#54812yes—
295C22×C53#55212yes—
296C22×C53#55612yes—
297C22×C53#55612yes—
298C22×C53#55912yes—
299C2×D53—42no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—82no—
309C2×D53—82no—
310C4×C53—24no—
311C4×C53—24no—
312C4×C53—28no—
313C4×C53—24no—
314C4×C53—44no—
315Dic53—44no—
316Dic53—44no—
317Dic53—42no—
318Dic53—81no—
319Dic53—81no—
320C22×C4×Q8D5311yescharacteristic, complemented
321C2×#393—531no—
322C2×#388—531no—
323C2×#388—532no—
324#1446—531no—
325#1442—531no—
326#1439—531no—
327#1437—532no—
328#1436—532no—
329#1436—532no—
330C4×#95—531no—
331C2×C53C4×#12011yescharacteristic
332C2×C53C4×#12411yescharacteristic
333C2×C53#232311yescharacteristic
334C2×C53#236212yes—
335C2×C53#236712yes—
336C2×C53#243011yescharacteristic
337C2×C53#243611yescharacteristic
338C2×C53#245812yes—
339C2×C53#246912yes—
340C2×C53#259211yescharacteristic
341C2×C53#260411yescharacteristic
342D53—82no—
343C23×Q8C2×D5311yescharacteristic
344C2×C4×Q8—28no—
345C2×C4×Q8—24no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C4⋊C4C2×D5312yes—
349C22×C22⋊C4—532no—
350C22×C22⋊C4—531no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—531no—
353C22×C22⋊C4—532no—
354C22×C42—532no—
355C22×C42C2×D5312yes—
356C22×C42C2×D5311yescharacteristic
357#393—538no—
358#388—1068no—
359#388—1064no—
360C2×#95—531no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95—531no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—531no—
370C2×#95—532no—
371C53#1418811yescharacteristic, complemented, Sylow
372C25—531no—
373C22×Q8—24no—
374C22×Q8—22no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8C4×D5314yes—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4C22×D5312yes—
384C23×C4C22×D5312yes—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C4×Q8—232no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4C22⋊D5318yes—
396C2×C22⋊C4—538no—
397C2×C22⋊C4—534no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—2122no—
409C2×C42—24no—
410C2×C42—24no—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—1064no—
414C2×C42—1064no—
415C2×C42—2122no—
416C2×C42C22⋊D5314yes—
417#95—534no—
418#95—538no—
419#95—1064no—
420#95—1062no—
421#95—1064no—
422#95—1064no—
423#95—1064no—
424#95—1064no—
425#95—1062no—
426#95—1062no—
427#95—2122no—
428C24—532no—
429C24—1062no—
430C24—1061no—
431C24—1062no—
432C24—1062no—
433C24—1061no—
434C24—1062no—
435C24—1061no—
436C24—1061no—
437C24—1061no—
438C24—1061no—
439C24C23×D5311yescharacteristic
440C2×Q8—28no—
441C2×Q8—216no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—216no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—532no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4C2×C4×D5314yes—
493C22×C4C2×C4×D5312yes—
494C22×C4#1171994514yes—
495C22×C4#1171994512yes—
496C22×C4#1171994914yes—
497C22×C4C2×C22⋊D5314yes—
498C22×C4C2×C22⋊D5314yes—
499C22×C4C2×C22⋊D5312yes—
500C22×C4C2×C22⋊D5312yes—
501C22×C4C2×C22⋊D5314yes—
502C22×C4C2×C22⋊D5312yes—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—2124no—
506C22⋊C4—1068no—
507C22⋊C4—10616no—
508C22⋊C4—2128no—
509C22⋊C4—2124no—
510C22⋊C4—2124no—
511C42—216no—
512C42—28no—
513C42—2128no—
514C23—1062no—
515C23—1064no—
516C23—1064no—
517C23—1062no—
518C23—1062no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23—2121no—
527C23—2122no—
528C23—2121no—
529C23—2121no—
530C23—2122no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2122no—
537C23C22×C4×D5311yescharacteristic
538C23C22×C22⋊D5312yes—
539C23C22×C22⋊D5311yescharacteristic
540C23#41×D5312yes—
541C23C2×#1171994512yes—
542C23C2×#1171994511yescharacteristic
543C23C2×#1171994912yes—
544C23#1313326311yescharacteristic
545C23#1313326411yescharacteristic
546C23#1313326511yescharacteristic
547C23#1313326511yescharacteristic
548Q8—232no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—1064no—
575C2×C4—1062no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4—2121no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4C4×C22⋊D5314yes—
604C2×C4C4×C22⋊D5318yes—
605C2×C4#1313318614yes—
606C2×C4#1313320614yes—
607C22—4242no—
608C22—4241no—
609C22—4241no—
610C22—4241no—
611C22—4242no—
612C22—4241no—
613C22—2122no—
614C22—2124no—
615C22—2122no—
616C22—2122no—
617C22—2124no—
618C22C2×C4×C22⋊D5311yescharacteristic
619C22C2×C4×C22⋊D5312yes—
620C22C4×#1171994912yes—
621C22#1681109911yescharacteristic
622C22#1681110511yescharacteristic
623C22C2×#1313318611yescharacteristic
624C22#1681113412yes—
625C22#1681113411yescharacteristic
626C22C2×#1313320611yescharacteristic
627C22#1681116012yes—
628C22#1681116911yescharacteristic
629C22#1681119711yescharacteristic
630C22#1681120111yescharacteristic
631C22#1681123111yescharacteristic
632C22#1681124511yescharacteristic
633C22#1681126112yes—
634C22#1681127212yes—
635C22#1681128412yes—
636C22#1681128412yes—
637C22#1681131412yes—
638C22#1681134512yes—
639C22#1681134612yes—
640C22#1681135812yes—
641C4—24no—
642C4—24no—
643C4—28no—
644C4—24no—
645C4—44no—
646C4—4241no—
647C4—4241no—
648C4—2122no—
649C4—2124no—
650C4—2124no—
651C2—4242no—
652C2C4×#1313318611yescharacteristic
653C2C4×#1313320611yescharacteristic
654C2#2027739111yescharacteristic
655C2#2027773812yes—
656C2#2027778212yes—
657C2#2027831911yescharacteristic
658C2#2027836911yescharacteristic
659C2#2027863112yes—
660C2#2027879012yes—
661C2#2028020311yescharacteristic
662C2#2028034711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C4×Q8×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC22×C4×Q8D53
hypocenterC53#14188
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20272986C2semidirect
5. #20272974C2semidirect
8. #20272948C2semidirect
9. #20272947C2semidirect
10. C4×#13133124C2semidirect
11. C22×C4×Q8×C53C2semidirect
19. #16810534C4semidirect
320. C22×C4×Q8D53semidirect
371. C53#14188semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168105342
C2×#168105142
C2×#168105132
#202729862
#202729742
#202729572
#202729532
#202729482
#202729472
C4×#131331242
C22×C4×Q8×C532
#1418853